New Developments in Reservoir Simulation and their Potential Impact on IO

Khalid Aziz

Stanford University

2

Define IO

Production System

Data

Update Detailed Model

Update Reduced Model

Optimization

Controls

Field Development Optimization

Optimization and

UncertaintyPropagation

Update Reduced Models

UpdateDetailed Models

Reservoir, Wells, Surface Facilities

New Wells or Facilities

3

Key Aspects of IO or Smart Fields

� Application of formal optimization techniques to

� Develop Field (wells, new wells, facilities)

� Including optimization while drilling

� Operate (control down hole and surface valves)

� All optimization requires assessing the impact of decisions

4

Assessing Impact of Decisions

� Reservoir Simulation

� Many thousands of simulations required

� Other Approximate Models

� Proxies (based on limited simulations)

� Huge potential of Formal Optimization

5

Development of Key Tools

� Optimization techniques (Smart Fields Consortium at Stanford - SFC)� Global or local

� Gradient based

� Stochastic

� Dynamic programming

� Multiobjective

� Simulation (Reservoir Simulation Consortium at Stanford – SUPRI-B)� Speed and robustness

� Upscaling and proxies

� Flexibility (new processes, updates, new hardware)

6

Modeling Environment

SimulatorEngine

PreprocessingPost-

Processing

Creating one or more images

Extracting useful results

Upscaling?

Wells andFacilities

7

Reservoir Simulation

( )110

i i i i i

w n nc c c c cR m m M M

t+= − − − =

∆

( )1

set of primary variables at iteration

Jacobian MatrixJ

J S S R

S

R

S

υ υ υ υ

υ

υυ

υ

υ

+ − = −

=

∂= =∂

r r r

%r

%

r

r

• Equations

From Gringarten

• Pressure drop / flow rate relationship

• Additional constraint equations

8

Each component is a module that hides its own functionality and data

Object Oriented Architecture

Younis 2011

9

Selected Simulation Research Reservoir Simulation Consortium at Stanford

1. Building a General Purpose Research Simulator (GPRS) using Automatic Differentiation (AD) to Build the Jacobian and Compute Gradients

2. A Robust Method for Solving Nonlinear Equations

3. Fast Flash Calculations for Complex Processes

Development of an Automatic

Differentiation Based GPRS

Work of Rami Younis and Yifan Zhou

PhD Students at Stanford University.

Additional contributions by other researchers.

11

AD (Automatic Differentiation)-GPRS: Motivation

• Need only nonlinear residual code

• Jacobian automatically generated and always accurate

• Flexible and extensible

• Need only nonlinear residual code

• Jacobian automatically generated and always accurate

• Flexible and extensible

AD Framework

• Avoid manual Jacobian construction

• Incorporate new physics

• Complex processes

• New formulations and solution algorithms

• Avoid manual Jacobian construction

• Incorporate new physics

• Complex processes

• New formulations and solution algorithms

GPRS using AD

• Flexible and efficient reservoir-simulation research laboratory

• Flexible and efficient reservoir-simulation research laboratory

Objectives of AD-GPRS

12

AD-GPRS: Key Features

� Generalized compositional formulation

� MPFA (Multi-Point Flux Approximation) discretization for unstructured grid

� Flexible multi-level AIM (Adaptive Implicit Method)

� Combination of MPFA and AIM

� General multi-segment wells

� ……

13

AD Framework

For each time-step

For each Newton iteration

RESIDUAL CODE

JACOBIAN CODE

ADscalarGradient

Value

double block_sparse_vector<block_size>

Automatically Generated JACOBIAN CODE

14

MPFA O(1/3)32:1

1

2

3

4

� 17,545 triangular cells with several linear no-flow features (mimic faults)

� 1 injector (center) + 4 producers (distributed around)

� Anisotropic ratio: 1:1, 2:1, 8:1, 32:1

� TPFA (4-pt), MPFA O(1/3)-method (18-pt)

TPFA32:1TPFA32:1

11

22

33

44

TPFA32:1

1

2

3

4

1

2

3

4

MPFA: Unstructured Grid1

MPFA – TPFA32:1

1 Y. Zhou, H. Tchelepi & B. Mallison, SPE 141592 (2011)

15

MPFA: Unstructured Grid

0

2

4

6

8

10

0 100 200 300 400 500 600

Ga

s R

ate

(k

m3/d

ay

)

Time (day)

1:1

MPFA P1

TPFA P10

2

4

6

8

10

0 100 200 300 400 500 600

Time (day)

2:1

0

2

4

6

8

10

0 100 200 300 400 500 600

Ga

s R

ate

(k

m3/d

ay

)

Time (day)

8:1

0

2

4

6

8

10

0 100 200 300 400 500 600

Time (day)

32:1

Zhou 2011

16

MPFA: Unstructured Grid Case

DiscretizationNewton

iterationsSolver

iterationsDiscretization

time (s)Solver time

(s)

Total time (s)

1:1TPFA

MPFA

597

576

9.5

5.6

0.08

0.18 (+114%)

0.33

0.60 (+82%)

0.61

0.98 (+60%)

2:1TPFA

MPFA

593

588

10.0

5.8

0.08

0.18 (+111%)

0.34

0.61 (+78%)

0.63

0.99 (+58%)

8:1TPFA

MPFA

573

581

11.6

10.2

0.09

0.18 (+113%)

0.40

0.85 (+110%)

0.69

1.24 (+78%)

32:1

TPFA

MPFA

602

639

12.3

20.8

0.09

0.19 (+96%)

0.45

1.62 (+261%)

0.77

2.02 (+161%)

� TPFA (4-pt), MPFA O(1/3)-method (18-pt, 3.5 times more)

Zhou 2011

17

Additional uses of ADGPRS

� Investigate selection of primary unknowns and equations

� Level of implicitness

� Different multipoint flux techniques

� Addition of new processes

� Chemical reactions

� Advanced wells

� …

Solving Nonlinear Equations

Work of Rami Younus

PhD Candidate at Stanford University

19

Implicit models and nonlinearity

( )1; , 0n nR S S t+ ∆ =

Challenges� Newton’s method may not converge

� Convergence rate may be too slow

� Time-step selection for convergence is hard

Try-Adapt-Try-Again strategy

Stiff Nonlinear Residual

( )11 1 1 1

01

; ,n n n n

n n

S S J R S S t

S S

ν ν ν

ν

++ + − +

=+

− = − ∆ =

Use Newton’s Method

20

Even ‘simple’ problems can be challenging

( ) ( )1 1n n ninjR S S c t f S f S+ + = − + ∆ −

2 phase incompressible flow in 1 cell

injS0

initS S=

1 ?S =

Dt∆

nS

1nS +

R

1nSν+

21

Generalized view of Newton’s iteration

( )1

1 1

1

0

; ,n

n n

n n

d SJ R S S t

d

S Sν

ν

+− +

+

=

= − ∆ =

Bottom line …� Newton’s iteration may not converge (unstable in

iteration index v)

� Derivation of ‘CFL-restriction-like’ criteria unlikely

Consider this IVP

It is stationary at the solution

( )11 1 1 1

01

; ,n n n n

n n

S S J R S S t

S S

ν ν ν

ν

ν++ + − +

=+

− = − ∆ =

∆Explicit Euler discretization yields Newton’s Iteration

Newton uses a step-size ∆ν=1

22

2 phase incompressible flow in 2 cells

1t∆ =

2S

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1S

( )1

2; ,n nR S S t+ ∆

Residual norm contours

4th order integration of Newton Flow

Ordinary Newton steps

injS1S 2S

23

‘Safe-guarding’ Newton’s Iteration (1)

( )11 1 1 1 ; ,n n n nS S J R S S tν ν ν

ν++ + − + − = − ∆ ∆

Classic methods select a constant� Standard: always 1

� Line-search: minimize residual norm along Newton direction

� Trust-region: minimize residual norm within a neighborhood of iterate

ν∆

24

‘Safe-guarding’ Newton’s Iteration (2)

Heuristics in commercial simulators � Appleyard Chop (AC) modifies Newton step cell-by-

cell:

� Changes from mobile to immobile are made barely mobile (and vice-versa)

� A count is kept to avoid ‘oscillation’

� EclipseTM Modified Appleyard Chop (MAC):

� Perform an Appleyard Chop AC

� Limit saturation changes by a maximum constant (0.2 works well)

25

Continuation-Newton

S

Time-step

target

0Sn

Select a step-size

Tangent from formulation

1

2

Fixed residual tolerance

Local Newton correction

3

26

Gravity Segregation

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

160

180

200

Time-step (days)

Num

ber

of It

erat

ions

No longer converges

No longer converges MAC Heuristic

(0.6)

MAC Heuristic (0.25)

CN – total iterates

CN – Newton corrections

27

Segregation and advection

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

0 50 100 1500

50

100

150

Time-step Size (Days)

Num

ber

of it

erat

ions

MAC Heuristic (0.6)

MAC Heuristic (0.25)

CN – total iterates

CN – Newton corrections

No longer converges

Fast Compositional Modeling

Work of D. Voskov, H. Pan, A. Iranshahr, R. Zaydullin.

Research Associates and Students

29

Tie-Simplex Parametrization

CO2 C10

p = const

T = const

zi = xi L + yi (1 - L)

z1 + z2 + z3 = 1

C4

iγjγz1

z3

z2

30

Why is CSAT efficient?

γ2

γ1

Solution invariance in γ-space

Solutions in one, two and three dimensions

31

Adaptive tabulation (CSAT)

p1< p2< p3< MCPp

• For given z & T, tabulate tie-lines for pressures up to the Minimal Critical Pressure (MCP)

• Construct refined table below MCPvapor liquid

p y1 y2 y3 x1 x2 x3

p1

p2

p3

MCP

p1 MCP

1 3

2

1 3

2

32

CSAT and RV performance

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Standard RV CSAT

Tim

e, s

ec

SIM time EoS time

SPE 3, immiscible gas injection SPE 5, miscible gas injection

33

Some other Areas of Current Research

� Modeling of Complex Wells

� Geomechanical Modeling

� Linear and Nonlinear Solvers

� Accurate Modeling of Capillary Heterogeneity

� Various Smart Fields Topics

34

Concluding Remarks

1. Simulator developments are making this technology more flexible, easily extendable, faster, and easier to develop new simulators

2. Academic and industrial collaborations have huge benefits for technology and human resource development

Thank You!

Questions or Comments?

Khalid AzizStanford University

36

SimulationCycle

UpdateDetailed Model

UpdateSimplified Model

Impose new Controls onPhysical System

Optimize

AssessValue of Objective

Function